Enumerating the number of specific patterns is another method to produce the count. However, it is supposed to take much more time than direct counting [1]}. For instance, when counting shortest cycles, one of the primary reasons is that the shortest cycle length is unknown in advance, and obtaining it requires the employment of a BFS-like method whose running time is already longer than the query time of hub labeling. Another reason is that enumeration requires finding all the vertices along with the cycles, which is unnecessary for the counting problem. With the help of a hot-point based index, [2]} can output the newly-generated constraint cycles upon the arrival of new edges. [3]} is shown to have better time performance than [2]} on enumerating constraint paths and cycles.
